- Email: [email protected]
The anomaly cancellation mechanism in N=1, D=4 supergravity and distorted supergravity algebra with Chern-Simon forms
Volume 169B, number 1
PHYSICS LETTERS
20 March 1986
T H E A N O M A L Y C A N C E L L A T I O N M E C H A N I S M IN N = 1, D = 4 S U P E R G R A V I T Y A N D D I S T O R T E D S U P E R G R A V I T Y ALGEBRA WITH C H E R N - S I M O N F O R M S Laurent B A U L I E U Laboratoire de Physique Thborique et Hautes Energies. Tour 16. ler btage. 4 Place Jussieu, F-75005 Paris. France
and Marc B E L L O N Laboratoire de Physique Thborique de FENS. 24 rue Lhomond, F-75005 Paris, France Received 25 October 1985
A mechanism is exhibited which ensures that N = 1, D = d new minimal supergravity is free of Lorentz× U(I) anomalies, for any coupling to matter, although it contains an abelian chiral gauge field. This is achieved through the determination of a new supergravity algebra characterized by the presence of a U ( I ) × Lorentz Chern-Simon form in the field strength of a two-form gauge field. Our analysis provides therefore an example in which Chern-Simon type interactions do occur, while preserving local supersymmetD'.
The possibility of an algebraic determination of off-shell transformation laws of supergravity, independently of a given invariant lagrangian, permits deeper analysis of anomalies in supergravity [1,2]. In this paper we solve a long-standing paradox which was believed to hold in N = 1, D = 4 Poincar6 supergravity, namely the possibility of anomalies in a given system of auxiliary fields, but not in other ones. We show that the a priori possible anomalies of the gauged chiral symmetry in the new minimal formulation o f N = I, D = 4 supergravity [3] are suppressed by a mechanism which is a new one although it has some similarity with the Green-Schwarz mechanism [4]. The specificity of this mechanism is that it does not rely on the introduction of a counterterm, but it nevertheless makes use of the presence of a two-form gauge field in the supergravity multiplet. A non-trivial result of our work is the proof that the mechanism is truly compatible with supersymmetry. The clue of our construction is the determination of a distorted N = 1, D = 4 closed system of supergravity transformations which involve Chern-Simon type couplings. Our analysis makes us confident that the anom0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
aly cancellation in N = 1, D = 10 supergravity, coupled to SO(32) or E(8) × E ( 8 ) , N = 1 super Yang-MiUs theory [4] is in fact compatible with local supersymmetry. In the new minimal formulation o f N = 1, D = 4 supergravity, the gauge fields are the vielbein e a = ea~ dx u, the spin-connection ~oab = ~ab u dx u, the gravitino xp = xI,u dx u, the auxiliary one-form abelian chiral gauge field A = A u dx u and the two-form gauge field B2 = ½Bu~,dx u d x v. T h e corresponding field strengths are T a = de a + wabeb +
½icpTaxP,
R =dw+co~, p = d~+ w~+AxP, F=dA, G 3 = dB 2 + ½i~Ta~e
a .
The gauge symmetry is known from the BRS operatots. Let I2 bb, X, B] and B02, c be the ghosts of the gauge fields ¢o, xp, B2, A, respectively, and/ju be the vector field ghost of diffeomorphisms. There is no 59
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need here to introduce either the anti-ghosts, or the anti-BRS operator. We define the modified ghosts as:
s~=s2 - i ~ ¢ o ,
20 March 1986
£ =f(¼eabcdeaebR cd + i~757ape a - 2BF + ~Ga~ G abe ~ eefgh eeefege h
X=x-i~, + matter field couplings).
[~ =B~ - i~B2 , B20 = B2 - iIB] + ~i~i~B2 , ~=c-i~A
,
in order to realize the now well-understood decoupling between internal symmetries and diffeomorphisms [1 ]. All ghost and classical objects are unified as follows: ~=e,
&=~+fi,
4,,=~+~,
/~ = &b +
which cannot be eliminated by means of a counterterm [2]
- d As(A, ~ o , F , g ) .
03 = aB2 + ½i~va~'e a ,
(1)
and ~=s-L~,
L~=i~d-di~.
(2)
The vector field with ghost number 2 c u is equal to -½i~,~u~. The action of ~, and thus the one of s, is determined consistently, i.e. with s2 = 0, from the following horizontality constraints, once they are expanded in ghost number [1 ]:
~.a = Ta = _½Gabc eb ec ' = p = ~Pabeae b , G3 = G3 = ~Gabc ea eb ec , --
(Sb)
= d ( F [ - 2 a A F - 2/3 T r ( t ~ - -~¢oww)]}
p=a,i,
¢]=d+§+i~,
(ba)
16 = - 2 ~ F F F - 2~F Tr (RR)
b = a,i, + ~,i, + j , i , ,
2i~.?[ayb]
(6)
tx and/3 are proportional to the gravity coupling constant and their values depend on the precise choice of matter fermions that are coupled. On the other hand, such anomalies are unlikely to occur really, since other systems of auxiliary fields exist which do not involve the chiral field A, and it is reasonable to expect that the physics described by supergravity should not depend on a particular choice of auxiliary fields. A Green-Schwarz type mechanism cannot be used however to discard these anomalies corresponding to eq. (6), because of the term B2F in the lagrangian (4), already present at the classical level. Rather we have found the following subtle mechanism. By setting = 0, the action becomes
f £~ ~ =f (~ eabcdeaebRCd -- 2BF --~
li X~.c~( Gab ! c
+ ~GabcGabc~ ee£gh eeefe geh
'
=F -{i~'757aYa - ~-~4ieabcdGbCd~Iaf(, with y a - iea p _ (Gabc ~ Obc + t~eabcdGbed 75 ) xp . and the s-invariant classical action is given by 60
~A~ + d A 2 = 0
At the bosonic level, i.e. with ~( = 0, one may fear pure abelian and mixed abelian and gravitational anomalies associated to the invariant six-form 16
~.a = ae a + &abeO + ~i@Ta~/'
t~ab = Rab
The possible anomalies are the solutions A1 of the Wess-Zumino consistency equation
+ sK ° + d r y .
/i=A +e,
(4)
(3)
+ matter field couplings).
(7)
The corresponding Feynman rules give the generating functional F tree of 1PI vertices, with sP tree = 0, under the condition that one chooses a BRS invariant gauge-fixing procedure. By computing the one-loop
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radiative corrections, with relevant renormalization conditions, one can determine p l loop. In turn, computing sP I loop, and due to the existence of I6 in eq. (6), one generally obtains that
sr 1loop (8) A t = ASQi, ¢3, F, R)I 1 = F [-2a~F - 2/3 Tr(fid¢o)l , where ~5 is defined in eq. (6). If A1 is not zero (a :~ 0 or/3 4= 0), the theory seems anomalous. Our observation, however, is that the anomaly is in fact spurious, even if the matter fields are chosen such that a or 3 is non-zero. The explanation is the possibility of distorting G3 into G3R G3R = dB2 + aA dA + 3 Tr(coR - ~cowco), dG3R = aFF +/3 Tr(RR).
(9)
The clue is that the Bianchi identity for G3R implies that one can define a modified BRS operator §R, together with the associated operators SR = SR + L~, dR and dR in a consistent way, i.e. with s2 = 0, in the absence of supersymmetry, by imposing the following horizontality conditions: (~3R = cIRB2 + oA dR.4 +/3 Tr(d0R - [ ~ G ~ ) = G3R,
TR=~IRe+~e=T, RR = a R ~ + r~& =R , FR = ~tRA = F .
which means that the model is anomaly-free independently of the value of a,/3, i.e. of the matter field content. The striking point of this procedure, which makes it different from the Green-Schwarz mechanism, is that it does not correspond only to the addition of a counterterm to the lagrangian, which contains in fact already a term B2F at the classical level. Rather, the coefficient of the anomaly appears within a distortion of the symmetry sR = s + a 6 s ,
(13)
and finally F satisfies relevant Ward identities, which leads to unitarity. We now extend this analysis to the case of the full supersymmetric theory, i.e. when ~ 4= 0. We will prove the existence of a modified BRS operator s R = s + a6 s which is a supersymmetric extension of SR defined by eq. (8). In fact, since we will consider one-loop results, we determine SR only at the order a, with the consistency condition s 2 = (s + o16 S)2 = O(a2) .
(14)
The supersymmetric solution of the anomaly consistency condition (5a), containing AI(O, 6, w, A) associated to A 5 in eq. (6), will be SR£ = eft6 s)£ with £ given by eq. (4) and G3 replaced by G3R. We have indeed from f s£ = 0 and f S2R£= O(a 2)
sfs¢.=f s2 .
2)
(10)
~R is identical to ~ up to terms of order a and/3 which
o cur in the
20 Match 1986
variations or82,
s02. With
new
symmetry we have AS = 2ctR(BJ~') -- 2G3R/7 •
(11)
This implies that A 1 in eq. (8) is equal to 2 [.~R(BF) +
d(&F)l and thus
S(SR£ ) + dA 2 = O(a2).
(15)
To determine SR, we compute the horizontality conditions on TR, RR, FR, PR, G3R which are compatible with the Bianchi identities RIR =/~R e
-- i~-~Ta/gR --
agkg=-[&ha], aRPR=0,
sgft(e,~,a,B,R,F, aR)=-fa14,
(12a)
sR r' 1 loop = O(a2,/32).
(12b)
Eq. (12b) proves that the computed F satisfies Ward identities at least up to the one-loop level. This fact, combined with the property s 2 = 0, is sufficient to prove that the one-loop physical S-matrix, formally defined from the LSZ reduction formula, is unitary
aa~R = (ka + FR) ,i, - ( ~ +A)A a , ~tR G3R = l i ~ 7 a ~ 7 ~
~ --
i~Ta~R ea + OtFR ~"R
+/3 T r ( R R k R ) .
(16)
To simplify our computations, we have worked in the case a = 8/3, but we are confident that ~R can as well be determined for any other value of a//3,1. ,1 For footnote see next page. 61
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We determine the relevant horizontality conditions perturbatively in/3. The lowest order solution is that of ref. [1 ] (eq. (3)). Inserting the lower order result in the Bianchi identity for GR, eq. (16), we obtain the order/3 result for G(R1) from the part with ghost number 3 G(R )= 4i/3~757dPf d,
fd -- leabcdGabc"
(17a)
This can be written as GR = ~ Gabc eaeb eC + 4i/3 ~ 7 5 " r a m
a
.
(17b)
From the part with ghost number 2 of the same Bianchi identity we determine p(l) and a new constraint for 7" ~aR -_ - - ~l nuab c e b.c , ~: =~ ¢°abc = Wabc(e, ~O) + ]Habc 1
1. d-i] 5 kl --~1/3"l Xede.i/klP "[ P
-- ~ i/375"),d~(ed eilkl~il p kl .
FR =~Fab ' ' e a e b --~l~O'y i. = 5 T a Pab , e b --¼i ~Taiha4¢ +/3e a e i i k l ( - ~1R i j gh Ogh +FijT5) Pkl] •
(19b)
We have verified that all other Bianchi identities remaining in eq. (16) are then satisfied with the new horizontality constraints (17)--(19). The whole set of BRS equations is determined in expanding in ghost number eqs. (17)-(19). Closure, i.e. s~t = 0, is guaranteed at least up to order t~. The gauge transformations follow from the SR-variation of classical fields. Up to order a, they read: SR ea = --½i~ "ra'4' -- ~ab eb '
SRXIt = - ( d + 60 + A ) X - (g), + c) Klff 1. d ^ -i] 5 kl e -fljpkl -- ~ff37 Xedei/klP "Y P -- ~i/3757a)(ed i/klta ,
~Habceaeb e c = ~Gabceaebe c + 4i/3~Td pdce c ,
p(l)=
20 March 1986
SRB 2 = - d / ~ l - i ~ 'a xPea - / 3 T r ( ~ d e ) (18a)
We define the quantity h a and the one-form 6 a by
- 8/36dA + 4i/3~T5"lapfd,
SRJ b = _(Off ab + [~, fil ab) + i~.rIa(o bl _ 2p'b]Ce e)
h _ 1. ubcd a - g eabcd *~ ) - ~ i edabc~7 d [2hc xI,
(18b)
I ' e aeb PR = Oa ea + ~Pab
•
_
1.
"
Oa--~l/37a~eiiklP
-i]
5
7 P
kl
1. 5 -i] kl --~1/37 7a~eijklP p .
1
(18c)
~ b = ½R,abcdeC e d _ 2i ~ 7 [ap'b]ce c + i ~7[a6b] ---~iedabc~TdthC~ 1 mn Omn + Fi/T 5) Pkl] +/3e c ei]kl(--~Ri]
mn
(rmn + Fi/'Y 5) Pkl] + i ~ 7 {aob] ,
, e b_¼i~(3,a[2ha~b SRA = - d c - ~ l X, . ± 7 5 3'a Pab
From the Bianchi identities for 7"R and PR we get finally the values of/~R and ~6R
(19a)
,l After completion of this work, our strong belief in the generalization of our results for a :~ B/8 has been comforted by a communication from Gixardi and Grimm, concerning the possibility of introducing Chern-Simon forms in superspace, with consistent constraints. 62
c
+ 1/3e ei/kl(-~Ri]
and
• a eiikl(--~Rii , mn Omn + Fii'[ 5) Pkt] + l[3e
(20)
We have defined o~ as the part with ghost number one of 6 a defined in (18c). There is no interest to display here the SR variations of ghosts. Observe that one still has SR~ ~t = ~ a t ~ ~ -- ½i~7u2. An interesting feature in the distorted supergravity algebra, eq. (20) is in particular that supersymmetry transforms the gravitino proportionally to the square of its field strength p. There is no difficulty to compute A14 from eq. (12a). The easiest way is to apply dR on the lagrangian (3) and to use the Bianchi identities (16). The resulting expression, which is awkward, contains the desired bosonic part/3 T r ( ~ &o) + 8/3~dA with all relevant supersymmetric counterparts. We have thus obtained two results. First, we have shown that it is possible to distort consistently the
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N = 1, D = 4 n e w minimal supergravity algebra in such a way that the field strength o f the t w o - f o r m contains the L o r e n t z and chiral C h e m - S i m o n s three-forms. The result is to the first non-trivial order in c~. It holds true presumably for all orders in ct but this w o u l d be o f no use for the a n o m a l y problem. Secondly, we have solved the long-standing p r o b l e m o f the chiral a n o m aly in the new minimal system of Sohnius and West, owing to a n e w m e c h a n i s m o f a n o m a l y cancellation which is m a d e possible by the existence o f the distorted supergravity algebra c o n s t r u c t e d in this article ,2. Our w o r k confirms the intuitive idea according to which different systems o f auxiliary field lead to the same on-shell physics. Let us conclude b y noticing
,2 We would like to suggest that a similar mechanism, based on building a distorted supergravity algebra, could solve another paradox found in N = 1, D -- I 1 supergravity, namely the generation of a Chern-Simon vertex through radiative corrections, in a way which seems to contradict the symmetry one starts from. The trick would be to express the theory with a six-form gauge field B6, dual to the three4ndex photon, with a field strength GTR = dB6 + aQ7 (w, R) rather than G~ = riB6, where Q7 is the Lorentz Chern-Simon form of rank 7.
20 March 1986
that, although our results have been established f r o m straightforward algebraic methods, their geometrical meaning is yet to be u n d e r s t o o d ,3 ,3 To simplify maximally the derivation of the distorted BRS constraints (17)-(19) by enforcing the compatibility of Bianchi identities and BRS symmetry, we have been naturally led to use the basis of one-forms (ea, ~) rather than the basis (ea, ~). The latter basis was used in ref. [1 ] to determine the undistorted BRS eqs. (3). The new basis (ea, ~) appears clearly as more natural in view of the identification of the gravitino as part of the supervierbein in superspace. On the other hand, it forces us to introduce field strengths with a prime symbol, such as Gabc m eq. (17b), Pab in eq. S t (18), R abcd and Fab in eq. (19), which differ from the field strengths deduced from the super-Poincat6 algebra, eq. (1), by gravitino-dependent terms.
References [1 ] L. Baulieu and M. Bellon, Phys. Lett. 161B (1985) 96; Nucl. Phys. B266 (1986) 75. [2] L. Baulieu, An introduction to supergravitational anomalies, LPTHE preprint 85/16, in: Proc. Anomaly Syrup. (Chicago, 1985), to be published. [3] M. Sohnius and P.C. West, Phys. Lett. 105B (1981) 353; Nucl. Phys. B198 (1982) 493. [4J M. Green and J. Schwarz, Phys. Lett. 149B (1984) 117; L. Baufieu, Anomaly evanescence and the occurrence of mixed abelian-nonabelian gauge symmetries, LPTHE preprint 85/09, Phys. Lett. B., to be published.
63
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20 March 1986
T H E A N O M A L Y C A N C E L L A T I O N M E C H A N I S M IN N = 1, D = 4 S U P E R G R A V I T Y A N D D I S T O R T E D S U P E R G R A V I T Y ALGEBRA WITH C H E R N - S I M O N F O R M S Laurent B A U L I E U Laboratoire de Physique Thborique et Hautes Energies. Tour 16. ler btage. 4 Place Jussieu, F-75005 Paris. France
and Marc B E L L O N Laboratoire de Physique Thborique de FENS. 24 rue Lhomond, F-75005 Paris, France Received 25 October 1985
A mechanism is exhibited which ensures that N = 1, D = d new minimal supergravity is free of Lorentz× U(I) anomalies, for any coupling to matter, although it contains an abelian chiral gauge field. This is achieved through the determination of a new supergravity algebra characterized by the presence of a U ( I ) × Lorentz Chern-Simon form in the field strength of a two-form gauge field. Our analysis provides therefore an example in which Chern-Simon type interactions do occur, while preserving local supersymmetD'.
The possibility of an algebraic determination of off-shell transformation laws of supergravity, independently of a given invariant lagrangian, permits deeper analysis of anomalies in supergravity [1,2]. In this paper we solve a long-standing paradox which was believed to hold in N = 1, D = 4 Poincar6 supergravity, namely the possibility of anomalies in a given system of auxiliary fields, but not in other ones. We show that the a priori possible anomalies of the gauged chiral symmetry in the new minimal formulation o f N = I, D = 4 supergravity [3] are suppressed by a mechanism which is a new one although it has some similarity with the Green-Schwarz mechanism [4]. The specificity of this mechanism is that it does not rely on the introduction of a counterterm, but it nevertheless makes use of the presence of a two-form gauge field in the supergravity multiplet. A non-trivial result of our work is the proof that the mechanism is truly compatible with supersymmetry. The clue of our construction is the determination of a distorted N = 1, D = 4 closed system of supergravity transformations which involve Chern-Simon type couplings. Our analysis makes us confident that the anom0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
aly cancellation in N = 1, D = 10 supergravity, coupled to SO(32) or E(8) × E ( 8 ) , N = 1 super Yang-MiUs theory [4] is in fact compatible with local supersymmetry. In the new minimal formulation o f N = 1, D = 4 supergravity, the gauge fields are the vielbein e a = ea~ dx u, the spin-connection ~oab = ~ab u dx u, the gravitino xp = xI,u dx u, the auxiliary one-form abelian chiral gauge field A = A u dx u and the two-form gauge field B2 = ½Bu~,dx u d x v. T h e corresponding field strengths are T a = de a + wabeb +
½icpTaxP,
R =dw+co~, p = d~+ w~+AxP, F=dA, G 3 = dB 2 + ½i~Ta~e
a .
The gauge symmetry is known from the BRS operatots. Let I2 bb, X, B] and B02, c be the ghosts of the gauge fields ¢o, xp, B2, A, respectively, and/ju be the vector field ghost of diffeomorphisms. There is no 59
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need here to introduce either the anti-ghosts, or the anti-BRS operator. We define the modified ghosts as:
s~=s2 - i ~ ¢ o ,
20 March 1986
£ =f(¼eabcdeaebR cd + i~757ape a - 2BF + ~Ga~ G abe ~ eefgh eeefege h
X=x-i~, + matter field couplings).
[~ =B~ - i~B2 , B20 = B2 - iIB] + ~i~i~B2 , ~=c-i~A
,
in order to realize the now well-understood decoupling between internal symmetries and diffeomorphisms [1 ]. All ghost and classical objects are unified as follows: ~=e,
&=~+fi,
4,,=~+~,
/~ = &b +
which cannot be eliminated by means of a counterterm [2]
- d As(A, ~ o , F , g ) .
03 = aB2 + ½i~va~'e a ,
(1)
and ~=s-L~,
L~=i~d-di~.
(2)
The vector field with ghost number 2 c u is equal to -½i~,~u~. The action of ~, and thus the one of s, is determined consistently, i.e. with s2 = 0, from the following horizontality constraints, once they are expanded in ghost number [1 ]:
~.a = Ta = _½Gabc eb ec ' = p = ~Pabeae b , G3 = G3 = ~Gabc ea eb ec , --
(Sb)
= d ( F [ - 2 a A F - 2/3 T r ( t ~ - -~¢oww)]}
p=a,i,
¢]=d+§+i~,
(ba)
16 = - 2 ~ F F F - 2~F Tr (RR)
b = a,i, + ~,i, + j , i , ,
2i~.?[ayb]
(6)
tx and/3 are proportional to the gravity coupling constant and their values depend on the precise choice of matter fermions that are coupled. On the other hand, such anomalies are unlikely to occur really, since other systems of auxiliary fields exist which do not involve the chiral field A, and it is reasonable to expect that the physics described by supergravity should not depend on a particular choice of auxiliary fields. A Green-Schwarz type mechanism cannot be used however to discard these anomalies corresponding to eq. (6), because of the term B2F in the lagrangian (4), already present at the classical level. Rather we have found the following subtle mechanism. By setting = 0, the action becomes
f £~ ~ =f (~ eabcdeaebRCd -- 2BF --~
li X~.c~( Gab ! c
+ ~GabcGabc~ ee£gh eeefe geh
'
=F -{i~'757aYa - ~-~4ieabcdGbCd~Iaf(, with y a - iea p _ (Gabc ~ Obc + t~eabcdGbed 75 ) xp . and the s-invariant classical action is given by 60
~A~ + d A 2 = 0
At the bosonic level, i.e. with ~( = 0, one may fear pure abelian and mixed abelian and gravitational anomalies associated to the invariant six-form 16
~.a = ae a + &abeO + ~i@Ta~/'
t~ab = Rab
The possible anomalies are the solutions A1 of the Wess-Zumino consistency equation
+ sK ° + d r y .
/i=A +e,
(4)
(3)
+ matter field couplings).
(7)
The corresponding Feynman rules give the generating functional F tree of 1PI vertices, with sP tree = 0, under the condition that one chooses a BRS invariant gauge-fixing procedure. By computing the one-loop
Volume 169B, number I
PHYSICS LETTERS
radiative corrections, with relevant renormalization conditions, one can determine p l loop. In turn, computing sP I loop, and due to the existence of I6 in eq. (6), one generally obtains that
sr 1loop (8) A t = ASQi, ¢3, F, R)I 1 = F [-2a~F - 2/3 Tr(fid¢o)l , where ~5 is defined in eq. (6). If A1 is not zero (a :~ 0 or/3 4= 0), the theory seems anomalous. Our observation, however, is that the anomaly is in fact spurious, even if the matter fields are chosen such that a or 3 is non-zero. The explanation is the possibility of distorting G3 into G3R G3R = dB2 + aA dA + 3 Tr(coR - ~cowco), dG3R = aFF +/3 Tr(RR).
(9)
The clue is that the Bianchi identity for G3R implies that one can define a modified BRS operator §R, together with the associated operators SR = SR + L~, dR and dR in a consistent way, i.e. with s2 = 0, in the absence of supersymmetry, by imposing the following horizontality conditions: (~3R = cIRB2 + oA dR.4 +/3 Tr(d0R - [ ~ G ~ ) = G3R,
TR=~IRe+~e=T, RR = a R ~ + r~& =R , FR = ~tRA = F .
which means that the model is anomaly-free independently of the value of a,/3, i.e. of the matter field content. The striking point of this procedure, which makes it different from the Green-Schwarz mechanism, is that it does not correspond only to the addition of a counterterm to the lagrangian, which contains in fact already a term B2F at the classical level. Rather, the coefficient of the anomaly appears within a distortion of the symmetry sR = s + a 6 s ,
(13)
and finally F satisfies relevant Ward identities, which leads to unitarity. We now extend this analysis to the case of the full supersymmetric theory, i.e. when ~ 4= 0. We will prove the existence of a modified BRS operator s R = s + a6 s which is a supersymmetric extension of SR defined by eq. (8). In fact, since we will consider one-loop results, we determine SR only at the order a, with the consistency condition s 2 = (s + o16 S)2 = O(a2) .
(14)
The supersymmetric solution of the anomaly consistency condition (5a), containing AI(O, 6, w, A) associated to A 5 in eq. (6), will be SR£ = eft6 s)£ with £ given by eq. (4) and G3 replaced by G3R. We have indeed from f s£ = 0 and f S2R£= O(a 2)
sfs¢.=f s2 .
2)
(10)
~R is identical to ~ up to terms of order a and/3 which
o cur in the
20 Match 1986
variations or82,
s02. With
new
symmetry we have AS = 2ctR(BJ~') -- 2G3R/7 •
(11)
This implies that A 1 in eq. (8) is equal to 2 [.~R(BF) +
d(&F)l and thus
S(SR£ ) + dA 2 = O(a2).
(15)
To determine SR, we compute the horizontality conditions on TR, RR, FR, PR, G3R which are compatible with the Bianchi identities RIR =/~R e
-- i~-~Ta/gR --
agkg=-[&ha], aRPR=0,
sgft(e,~,a,B,R,F, aR)=-fa14,
(12a)
sR r' 1 loop = O(a2,/32).
(12b)
Eq. (12b) proves that the computed F satisfies Ward identities at least up to the one-loop level. This fact, combined with the property s 2 = 0, is sufficient to prove that the one-loop physical S-matrix, formally defined from the LSZ reduction formula, is unitary
aa~R = (ka + FR) ,i, - ( ~ +A)A a , ~tR G3R = l i ~ 7 a ~ 7 ~
~ --
i~Ta~R ea + OtFR ~"R
+/3 T r ( R R k R ) .
(16)
To simplify our computations, we have worked in the case a = 8/3, but we are confident that ~R can as well be determined for any other value of a//3,1. ,1 For footnote see next page. 61
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We determine the relevant horizontality conditions perturbatively in/3. The lowest order solution is that of ref. [1 ] (eq. (3)). Inserting the lower order result in the Bianchi identity for GR, eq. (16), we obtain the order/3 result for G(R1) from the part with ghost number 3 G(R )= 4i/3~757dPf d,
fd -- leabcdGabc"
(17a)
This can be written as GR = ~ Gabc eaeb eC + 4i/3 ~ 7 5 " r a m
a
.
(17b)
From the part with ghost number 2 of the same Bianchi identity we determine p(l) and a new constraint for 7" ~aR -_ - - ~l nuab c e b.c , ~: =~ ¢°abc = Wabc(e, ~O) + ]Habc 1
1. d-i] 5 kl --~1/3"l Xede.i/klP "[ P
-- ~ i/375"),d~(ed eilkl~il p kl .
FR =~Fab ' ' e a e b --~l~O'y i. = 5 T a Pab , e b --¼i ~Taiha4¢ +/3e a e i i k l ( - ~1R i j gh Ogh +FijT5) Pkl] •
(19b)
We have verified that all other Bianchi identities remaining in eq. (16) are then satisfied with the new horizontality constraints (17)--(19). The whole set of BRS equations is determined in expanding in ghost number eqs. (17)-(19). Closure, i.e. s~t = 0, is guaranteed at least up to order t~. The gauge transformations follow from the SR-variation of classical fields. Up to order a, they read: SR ea = --½i~ "ra'4' -- ~ab eb '
SRXIt = - ( d + 60 + A ) X - (g), + c) Klff 1. d ^ -i] 5 kl e -fljpkl -- ~ff37 Xedei/klP "Y P -- ~i/3757a)(ed i/klta ,
~Habceaeb e c = ~Gabceaebe c + 4i/3~Td pdce c ,
p(l)=
20 March 1986
SRB 2 = - d / ~ l - i ~ 'a xPea - / 3 T r ( ~ d e ) (18a)
We define the quantity h a and the one-form 6 a by
- 8/36dA + 4i/3~T5"lapfd,
SRJ b = _(Off ab + [~, fil ab) + i~.rIa(o bl _ 2p'b]Ce e)
h _ 1. ubcd a - g eabcd *~ ) - ~ i edabc~7 d [2hc xI,
(18b)
I ' e aeb PR = Oa ea + ~Pab
•
_
1.
"
Oa--~l/37a~eiiklP
-i]
5
7 P
kl
1. 5 -i] kl --~1/37 7a~eijklP p .
1
(18c)
~ b = ½R,abcdeC e d _ 2i ~ 7 [ap'b]ce c + i ~7[a6b] ---~iedabc~TdthC~ 1 mn Omn + Fi/T 5) Pkl] +/3e c ei]kl(--~Ri]
mn
(rmn + Fi/'Y 5) Pkl] + i ~ 7 {aob] ,
, e b_¼i~(3,a[2ha~b SRA = - d c - ~ l X, . ± 7 5 3'a Pab
From the Bianchi identities for 7"R and PR we get finally the values of/~R and ~6R
(19a)
,l After completion of this work, our strong belief in the generalization of our results for a :~ B/8 has been comforted by a communication from Gixardi and Grimm, concerning the possibility of introducing Chern-Simon forms in superspace, with consistent constraints. 62
c
+ 1/3e ei/kl(-~Ri]
and
• a eiikl(--~Rii , mn Omn + Fii'[ 5) Pkt] + l[3e
(20)
We have defined o~ as the part with ghost number one of 6 a defined in (18c). There is no interest to display here the SR variations of ghosts. Observe that one still has SR~ ~t = ~ a t ~ ~ -- ½i~7u2. An interesting feature in the distorted supergravity algebra, eq. (20) is in particular that supersymmetry transforms the gravitino proportionally to the square of its field strength p. There is no difficulty to compute A14 from eq. (12a). The easiest way is to apply dR on the lagrangian (3) and to use the Bianchi identities (16). The resulting expression, which is awkward, contains the desired bosonic part/3 T r ( ~ &o) + 8/3~dA with all relevant supersymmetric counterparts. We have thus obtained two results. First, we have shown that it is possible to distort consistently the
Volume 169B, number 1
PHYSICS LETTERS
N = 1, D = 4 n e w minimal supergravity algebra in such a way that the field strength o f the t w o - f o r m contains the L o r e n t z and chiral C h e m - S i m o n s three-forms. The result is to the first non-trivial order in c~. It holds true presumably for all orders in ct but this w o u l d be o f no use for the a n o m a l y problem. Secondly, we have solved the long-standing p r o b l e m o f the chiral a n o m aly in the new minimal system of Sohnius and West, owing to a n e w m e c h a n i s m o f a n o m a l y cancellation which is m a d e possible by the existence o f the distorted supergravity algebra c o n s t r u c t e d in this article ,2. Our w o r k confirms the intuitive idea according to which different systems o f auxiliary field lead to the same on-shell physics. Let us conclude b y noticing
,2 We would like to suggest that a similar mechanism, based on building a distorted supergravity algebra, could solve another paradox found in N = 1, D -- I 1 supergravity, namely the generation of a Chern-Simon vertex through radiative corrections, in a way which seems to contradict the symmetry one starts from. The trick would be to express the theory with a six-form gauge field B6, dual to the three4ndex photon, with a field strength GTR = dB6 + aQ7 (w, R) rather than G~ = riB6, where Q7 is the Lorentz Chern-Simon form of rank 7.
20 March 1986
that, although our results have been established f r o m straightforward algebraic methods, their geometrical meaning is yet to be u n d e r s t o o d ,3 ,3 To simplify maximally the derivation of the distorted BRS constraints (17)-(19) by enforcing the compatibility of Bianchi identities and BRS symmetry, we have been naturally led to use the basis of one-forms (ea, ~) rather than the basis (ea, ~). The latter basis was used in ref. [1 ] to determine the undistorted BRS eqs. (3). The new basis (ea, ~) appears clearly as more natural in view of the identification of the gravitino as part of the supervierbein in superspace. On the other hand, it forces us to introduce field strengths with a prime symbol, such as Gabc m eq. (17b), Pab in eq. S t (18), R abcd and Fab in eq. (19), which differ from the field strengths deduced from the super-Poincat6 algebra, eq. (1), by gravitino-dependent terms.
References [1 ] L. Baulieu and M. Bellon, Phys. Lett. 161B (1985) 96; Nucl. Phys. B266 (1986) 75. [2] L. Baulieu, An introduction to supergravitational anomalies, LPTHE preprint 85/16, in: Proc. Anomaly Syrup. (Chicago, 1985), to be published. [3] M. Sohnius and P.C. West, Phys. Lett. 105B (1981) 353; Nucl. Phys. B198 (1982) 493. [4J M. Green and J. Schwarz, Phys. Lett. 149B (1984) 117; L. Baufieu, Anomaly evanescence and the occurrence of mixed abelian-nonabelian gauge symmetries, LPTHE preprint 85/09, Phys. Lett. B., to be published.
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